As we have seen, vector addition and scalar multiplication
can produce new vectors out of old ones. For instance, we produce
the vector A + B by
adding the two vectors A and B.
Of course, there is nothing that makes A + B at
all distinct as a vector from A or B:
all three have magnitudes and directions. And just as A + B can
be construed as the sum of two other vectors, so can A and B.
In problems involving vector addition, it’s often convenient to
break a vector down into two components, that is, two
vectors whose sum is the vector in question.

Basis Vectors

We often graph vectors in an xy-coordinate
system, where we can talk about vectors in purely numerical terms.
For instance, the vector (3,4) is the vector whose
tail is at the origin and whose tip is at the point (3,4) on
the coordinate plane. From this coordinate, you can use the Pythagorean
Theorem to calculate that the vector’s magnitude is 5 and
trigonometry to calculate that its direction is about 53.1º above
the x-axis.

Two vectors of particular note are (1,0), the
vector of magnitude 1 that points along the x-axis,
and (0,1), the vector of magnitude 1 that
points along the y-axis. These are called the basis
vectors and are written with the special hat notation: and respectively.

The basis vectors are important because you can express
any vector in terms of the sum of multiples of the two basis vectors.
For instance, the vector (3,4) that we discussed above—call
it A—can be expressed
as the vector sum .

The vector is called the “x-component”
of A and the is
called the “y-component” of A.
In this book, we will use subscripts to denote vector components.
For example, the x-component of A is and
the y-component of vector A is .

The direction of a vector can be expressed in terms of
the angle by which it is rotated counterclockwise
from the x-axis.

Vector Decomposition

The process of finding a vector’s components is known
as “resolving,” “decomposing,” or “breaking down” a vector. Let’s
take the example, illustrated above, of a vector, A,
with a magnitude of A and a direction above
the x-axis. Because , ,
and A form a right triangle,
we can use trigonometry to solve this problem. Applying the trigonometric
definitions of cosine and sine,

we find:

Vector Addition Using Components

Vector decomposition is particularly useful when you’re
called upon to add two vectors that are neither parallel nor perpendicular.
In such a case, you will want to resolve one vector into components
that run parallel and perpendicular to the other vector.

Example

Two
ropes are tied to a box on a frictionless surface. One rope pulls
due east with a force of 2.0N. The second rope pulls with a force
of 4.0N at an angle 30Âº west of north, as shown in the diagram.
What is the total force acting on the box?

To solve this problem, we need to resolve the force on
the second rope into its northward and westward components.

Because the force is directed 30º west of north, its northward
component is

and its westward component is

Since the eastward component is also 2.0N,
the eastward and westward components cancel one another out. The
resultant force is directed due north, with a force of approximately 3.4N.

You can justify this answer by using the parallelogram
method. If you fill out the half-completed parallelogram formed
by the two vectors in the diagram above, you will find that the
opposite corner of the parallelogram is directly above the corner
made by the tails of those two vectors.